Differentiation, derivatives and Taylor approximations: Tangent line
Derivative of a function
Below, the graph of a smooth function has been plotted together with the tangent line at the point . In the immediate neighbourhood of this point, the graph and the tangent line can hardly be distinguished. As an illustration of this, the marked rectangle around the point in the figure below is displayed in an enlarged format.
In the below interactive diagram you can explore this behaviour. Yan can drag the point over the graph and zoom in on the situation around this point by making the marked square around the point smaller: the drawn tangent line and the graph of the function are then more similar.
For a smooth function it is true that when gets close to a fixed the following approximation is valid:
The equation of the tangent line at the point is equal to
In the interactive diagram to the right you can drag the point over the graph and observe the tangent line at that point.
The above statement also means that when the function value and derivatives for a certain are known, one also can estimate the function values at a distance with the following formula:
For a smooth function over an interval there is at any point a derivative over that interval defined:
Let us calculate the derivative of the polynomial function of degree 2 from a previous example.
For the polynomial function we calculate the difference quotient over the interval .
Computing a derivative in the above manner is cumbersome, but fortunately derivatives of many standard functions is listed and there exist calculation rules to differentiate combinations of functions efficiently. An example of such rule is the constant factor rule voor differentiation a multiple of a function for which the derivative is known.
Constant factor rule If is a function of which the derivative is known, then we have:
Mathcentre video
Differentiation First Principles - Tangent (30:44)